On the Fourier Transform of Bessel Functions over Complex Numbers---I: the Spherical Case

In this note, we shall prove a formula for the Fourier transform of spherical Bessel functions over complex numbers, viewed as the complex analogue of the classical formulae of Hardy and Weber. The formula has strong representation theoretic motivations in the Waldspurger correspondence over the complex field.Comment: 7 page

On the Fourier Transform of Bessel Functions over Complex Numbers---II: the General Case

In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura-Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields.Comment: 25 pages, to appear in Trans. Amer. Math. So

On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields

In this article, we prove certain Weber-Schafheitlin type integral formulae for Bessel functions over complex numbers. A special case is a formula for the Fourier transform of regularized Bessel functions on complex numbers. This is applied to extend the work of A. Venkatesh on Beyond Endoscopy for $\mathrm{Sym}^2$ on $\mathrm{GL}_2$ from totally real to arbitrary number fields.Comment: 26 pages. An appendix added. Small errors corrected. To appear in IMR

On the Kuznetsov Trace Formula for PGL2(C)\mathrm{PGL}_2(\mathbb{C})

In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group $\Gamma$ in $\mathrm{PGL}_2(\mathbb{C})$ that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for $\mathrm{PGL}_2(\mathbb{C})$. This approach avoids the difficult analysis in the existing method due to Bruggeman and Motohashi.Comment: 19 pages. J. Funct. Anal. Some calculations and statements for complementary series are simplified (corrected), in which case d should be 0 (not an arbitrary integer

A Whittaker-Plancherel Inversion Formula for SL2(C)\mathrm{SL}_2(\mathbb{C})

In this paper, we establish a Whittaker-Plancherel inversion formula for $\mathrm{SL}_2(\mathbb{C})$ from the analytic perspective of the Bessel transform of Bruggeman and Motohashi. The formula gives a decomposition of the Whittaker-Fourier coefficient of a compactly supported function on $\mathrm{SL}_2(\mathbb{C})$ in terms of its Bessel coefficients attached to irreducible unitary tempered representations of $\mathrm{SL}_2(\mathbb{C})$.Comment: 17 pages. Analysis greatly simplified thanks to an observation of the referee. To appear in J. Anal. Mat

Morita's Theory for the Symplectic Groups

We construct and study the holomorphic discrete series representation and the principal series representation of the symplectic group $\mathrm{Sp}(2n,F)$ over a $p$-adic field $F$ as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for $\mathrm{SL}(2, F)$ defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.Comment: 23 page

Learning to Confuse: Generating Training Time Adversarial Data with Auto-Encoder

In this work, we consider one challenging training time attack by modifying training data with bounded perturbation, hoping to manipulate the behavior (both targeted or non-targeted) of any corresponding trained classifier during test time when facing clean samples. To achieve this, we proposed to use an auto-encoder-like network to generate the pertubation on the training data paired with one differentiable system acting as the imaginary victim classifier. The perturbation generator will learn to update its weights by watching the training procedure of the imaginary classifier in order to produce the most harmful and imperceivable noise which in turn will lead the lowest generalization power for the victim classifier. This can be formulated into a non-linear equality constrained optimization problem. Unlike GANs, solving such problem is computationally challenging, we then proposed a simple yet effective procedure to decouple the alternating updates for the two networks for stability. The method proposed in this paper can be easily extended to the label specific setting where the attacker can manipulate the predictions of the victim classifiers according to some predefined rules rather than only making wrong predictions. Experiments on various datasets including CIFAR-10 and a reduced version of ImageNet confirmed the effectiveness of the proposed method and empirical results showed that, such bounded perturbation have good transferability regardless of which classifier the victim is actually using on image data

Hybrid subconvexity bounds for L(12,Sym2f⊗g)L \left(\tfrac{1}{2}, \text{Sym}^2 f \otimes g\right)

Fix an integer $\kappa\geqslant 2$. Let $P$ be prime and let $k> \kappa$ be an even integer. For $f$ a holomorphic cusp form of weight $k$ and full level and $g$ a primitive holomorphic cusp form of weight $2 \kappa$ and level $P$, we prove hybrid subconvexity bounds for $L \left(\tfrac{1}{2}, \text{Sym}^2 f \otimes g\right)$ in the $k$ and $P$ aspects when $P^{\frac {13} {64} + \delta} < k < P^{\frac 3 8 - \delta}$ for any $0 < \delta < \frac {11} {128}$. These bounds are achieved through a first moment method (with amplification when $P^{\frac {13} {64}} < k \leqslant P^{\frac 4 {13}}$).Comment: 1st draft, 27 page

The time evolution of coherent atomic system and probe light in an EIT medium

The adiabatic solutions of Maxwell-Bloch equation governing the three-level EIT medium is presented. The time evolution of the density matrix elements of the EIT system and the probe light is thus investigated by using the adiabatic approximation formulation and the slowly varying envelope condition.Comment: 6 pages, Latex, one figur

Maintenance of Coherence and Polarization Evolution in a Supersymmetric Multiphoton Model

The elimination of decoherence of two-state quantum systems interacting with a thermal reservoir through an external controllable driving field is discussed in the present paper. The restriction equation with which the external controllable driving field should agree will be derived. Based on this, we obtain the time-development equation of the off-diagonal elements of density operator in the supersymmetric multiphoton two-state quantum systems, which is helpful for studying the polarization evolution in this two-state quantum model.Comment: 5 pages, Late